If you had to shorten your LO's nighttime sleep because of daycare did they eventually compensate with a longer nap?

E currently sleeps 12.5-13 hours a night, but when I start work in early April I think I'll need to shorten it. I'm just not sure how I can manage to do everything that we need to do with the current amount of sleep he's currently. They serve breakfast at the daycare so I can probably skip that at home, but we'll have a commute. Then at night another commute, plus dinner, bath, reading, etc.

I'd like to be able to push his bedtime back so we can have a little bit of time together in the evening that isn't rushed, and I'm hoping he'll make up for it with a longer nap. Or am I deluding myself?